You can delete individual photos from the photos page, as well as from Photostream on your Flickr account. To delete photos from photo page, open the photo which you want to delete. Tap 'More' menu located at the bottom right corner of the page and select 'Delete this photo' option.
If you want to delete photos from Photostream, then go to your Photostream and click 'Edit' tab located above the photos. Click 'Delete' option located under the respective photo. When you click 'Delete' option, it will ask you 'Are you sure you want to delete this photo?'. Click 'Yes, Delete it' button to confirm it. If you want to delete multiple photos at a time, then gather them into a batch and perform a batch deletion. If you want to gather them into a batch, then tag photos with 'x' or select photos from a certain date.
Quick Tip to ensure your Photos never go missing:
If you are thinking of moving away from Flickr and to any other online location to keep your photos safely backed up, then you may need PicBackMan Flickr Uploader to easily move/transfer photos from Flickr to any other supported service of your choice. Simply download PicBackMan (it's free!), register your account, connect to your online accounts and tell PicBackMan where your photos are - PicBackMan does the rest, automatically.